I'm level 43, and have 8 Legendaries, 5 level , and 22 level fighters and have never won battle of the day, or any level up materials there, not even once. All it has ever done is drain my stamina. I go in as heavy as I can, with color variety.
I can win everywhere else.

The 'Battle of the Day' will typically be the Material Island featured on that day, such as Monday Seeds, Tuesday Mist, or Friday Nest. On the week-end, the Battle of the Day will be the featured Swamp that day. For the Daily Mission of 'Win Battle of the Day,' you will need to play that Island & win. You may play on any Tier of the Island (1-Sword, 2-Sword, or 3-Sword), but you will have to win the match to successfully pass the Daily Mission.

Many of the Evo Materials found on the weekly Islands have a high armour rating. The simplest approach to winning these Islands is using Captain Fighters that can ignore armour, like Anubis or Hel or even Xoniac. You can also use Fighters like Odin, Minotaur, Torikhan, or Cancer to ignore enemy armour for an amount of turns. Another option is poison. There are many Fighters that can inflict poison damage, including Duo Tikki & Spectres. There are also a few Fighters with Active Skill attacks that can penetrate enemy armour, like the Sharpshooters (Drake, Naurim, Grimer, etc.), Kali, Wrath, & Hera. Then there are Fighters like the Orcs that can reduce enemy shield by a percentage, making it easier to surpass their armour rating.

If none of thse options are available to you, you will need to take the high-attack route. You will want to use attack-boosting Captains &/or Fighters with Active Skills that boost attack. To deal damage to an enemy your attacks will need to surpass the enemy's armour. For example: A Marbled Jewelry Chest has an armour rating of 25.000. This means that to deal damage to it, your attacks will need to be higher than 25.000.

I hope that helps! If you would be comfortable with sharing your Fighters here, perhaps we can give you some advice on what Fighters to use on your team to win these battles.